The teeth on your cassette might be broken but this is pretty uncommon. You might be seeing the machined teeth that are designed to allow the chain to move easily from cog to cog. To those unfamiliar with cassettes, they can look broken, but if they are smooth instead of jagged it's likely they are supposed to be that way.
Shimano and SRAM are perhaps the two most common manufacturers of cassettes. Sunrace makes cassettes too, along with some other companies. For the most part cassette quality determines weight and how much of that shift-assisting machining there is that I mentioned above. Most cassettes are primarily made of steel, with various coatings to prevent rust. High-end stuff will have titanium or aluminum cogs to help eliminate weight.
It's a pretty big jump from a 32 down to a 26. The rear derailleur will likely work but if it doesn't work well it's possible there's some incompatibility going on with the rear der. That's more of a road bike gear range.